Images of Galois representations in mod $p$ Hecke algebras

Let $(\mathbb{T}_f,\mathfrak{m}_f)$ denote the mod $p$ local Hecke algebra attached to a normalised Hecke eigenform $f$, which is a commutative algebra over some finite field $\mathbb{F}_q$ of characteristic $p$ and with residue field $\mathbb{F}_q$. By a result of Carayol we know that, if the residual Galois representation $\overline{\rho}_f:G_\mathbb{Q}\rightarrow\mathrm{GL}_2(\mathbb{F}_q)$ is absolutely irreducible, then one can attach to this algebra a Galois representation $\rho_f:G_\mathbb{Q}\rightarrow\mathrm{GL}_2(\mathbb{T}_f)$ that is a lift of $\overline{\rho}_f$. We will show how one can determine the image of $\rho_f$ under the assumptions that $(i)$ the image of the residual representation contains $\mathrm{SL}_2(\mathbb{F}_q)$, $(ii)$ that $\mathfrak{m}_f^2=0$ and $(iii)$ that the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow us to deduce the existence of certain $p$-elementary abelian extensions of big non-solvable number fields.